Question 1 :¶
import numpy as np
import numpy.random as sim
import matplotlib.pyplot as plt
import matplotlib
matplotlib.rcParams['figure.figsize'] = (20.0, 10.0)
n =1000
r0 =3/100
alpha = 0.1
beta =1/100
theta = 5/100
T=5
pas = T/n
N=5
r=np.ones((n+1,N))*r0
for j in range(N):
for i in range(1,n+1):
deltaWt = np.sqrt(pas)*sim.randn()
r[i,j] = r[i-1,j] + alpha*(beta-r[i-1,j])*pas+ theta*deltaWt
dates=np.linspace(0,T,n+1) # n+1 dates
graph=plt.plot(dates,r)
plt.show()
n =1000
r0 =3/100
alpha = 0.1
beta =1/100
theta = 5/100
T=5
pas = T/n
N=5
r=np.ones((n+1,N))*r0
integ=np.zeros((n+1,N))*r0
for j in range(N):
for i in range(1,n+1):
deltaWt = np.sqrt(pas)*sim.randn()
r[i,j] = r[i-1,j] + alpha*(beta-r[i-1,j])*pas+ theta*deltaWt
integ[i,j] = pas* np.sum(r[0:i-1,j]) #ici on a calculé l'intégrale pour chaque trajectoire
print(integ)
[[ 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00] [ 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00] [ 1.50000000e-04 1.50000000e-04 1.50000000e-04 1.50000000e-04 1.50000000e-04] ... [-2.66025954e-01 -1.66335087e-01 -7.92488926e-02 2.42862488e-01 2.53967123e-01] [-2.66571362e-01 -1.66422262e-01 -7.93945190e-02 2.43004285e-01 2.53719508e-01] [-2.67137784e-01 -1.66519203e-01 -7.95460164e-02 2.43114941e-01 2.53486992e-01]]
n =1000
r0 =3/100
alpha = 0.1
beta =1/100
theta = 5/100
T=5
pas = T/n
N=5
r=np.ones((n+1,N))*r0
integ=np.zeros((n+1,N))*r0
Bt=[1] # le vecteur qui stock les Bt(i,n)
R=[]
CoeffActualisation= np.ones((n+1,N)) # Cette variable sert a stocké l'esperence de l'expenentiel de l'intégrale pour ensuite, calculer ça moyenne
for j in range(N):
for i in range(1,n+1):
deltaWt = np.sqrt(pas)*sim.randn()
r[i,j] = r[i-1,j] + alpha*(beta-r[i-1,j])*pas+ theta*deltaWt
integ[i,j] = pas* np.sum(r[0:i-1,j]) #ici on a calculé l'intégrale pour chaque trajectoire
CoeffActualisation[i,j] = np.exp(-integ[i,j])
# 2eme etape sommer les Bt(i,n)
for i in range(1,n+1):
moyenneBt=np.mean(CoeffActualisation[i,:])
Bt.append(moyenneBt)
R.append(moyenneBt**(-1/(pas*i))-1)
R.append(R[-1])
# le vecteur ou on va stocké les valeur du taux Zc R(t)
#on a mis une valeur a cause du problème de n+1 car la longeur de R est n et le rajout de cette valeure sert a la représentation
dates=np.linspace(0,T,n+1) # n+1 dates
graph=plt.plot(dates,R)
plt.show()
print("les prix du B0,T sont : ")
print(Bt)
les prix du B0,T sont : [1, 1.0, 0.9998500112494375, 0.9996836412954458, 0.9995111987501784, 0.9993440753710885, 0.9991868665838941, 0.9990225990150771, 0.9988527749673641, 0.9986878722528838, 0.9985250781807806, 0.9983616242289471, 0.9981835166857371, 0.9980135209750587, 0.99783074686625, 0.9976462341383392, 0.9974672654964989, 0.9972804772041612, 0.9971069776273389, 0.9969281510010063, 0.9967427806592302, 0.9965716380951763, 0.9963992780745425, 0.9962352989085475, 0.9960757622833187, 0.995916464557645, 0.9957480384746509, 0.9955695317444688, 0.9954012531422327, 0.9952403709943128, 0.9950805512736685, 0.9949239735878536, 0.9947742392730516, 0.9946230011627659, 0.9944801736430662, 0.9943349019832135, 0.9941889604020826, 0.9940419368352362, 0.9938977649610736, 0.993745315308004, 0.993589907437299, 0.9934383330398762, 0.9932817765856375, 0.9931316603848716, 0.992985186719398, 0.9928278017378611, 0.9926719939532319, 0.9925249383087345, 0.9923657975002941, 0.9922292861196599, 0.9920969940922406, 0.9919545807051765, 0.9918117719418195, 0.9916736625944148, 0.9915356671676621, 0.9913940848542862, 0.991250341545266, 0.9911068781051917, 0.9909581829092076, 0.9908042009202355, 0.9906572958382387, 0.990511795046108, 0.9903715361149775, 0.9902419759772766, 0.9901120038152074, 0.9899821035117421, 0.9898565840431669, 0.9897323676659783, 0.9896010786562123, 0.9894687015990714, 0.9893322139893733, 0.9891900386755765, 0.9890593682046613, 0.9889258262191639, 0.9887909815855433, 0.9886605613802708, 0.9885219254957383, 0.9883869768905209, 0.9882487022887417, 0.9881072607718115, 0.9879718705581281, 0.9878251529615166, 0.9876669103907731, 0.9875072275977217, 0.9873407087152376, 0.9871702034643723, 0.9869936574578455, 0.9868126808968792, 0.9866373420581475, 0.9864658547928533, 0.9862998716884673, 0.9861409135693819, 0.9859897534087875, 0.9858362255617694, 0.9856817557224767, 0.9855306478942613, 0.9853686120878222, 0.9851937359782758, 0.9850137471521702, 0.9848383705974353, 0.9846660363991301, 0.9844965517239302, 0.9843437861028065, 0.9841886760809924, 0.9840515564169975, 0.9839087325211174, 0.983753287895837, 0.9836096191392262, 0.9834584028979672, 0.9833071063637859, 0.9831658797543714, 0.9830274769789282, 0.9828848782743845, 0.9827480591486729, 0.9825987645081602, 0.9824594577588954, 0.9823242475060405, 0.9821757551293976, 0.9820218431053547, 0.9818567718655666, 0.9816817859502077, 0.9815171034950311, 0.9813396340642454, 0.9811761077242858, 0.9810065940419538, 0.9808440445688069, 0.9806832879274727, 0.9805109762576603, 0.9803335391530899, 0.9801490479003434, 0.9799656170958937, 0.9797909908538562, 0.9796295302131484, 0.9794676945304985, 0.9793108130357091, 0.9791497739869026, 0.9789838536427432, 0.9788243600942955, 0.9786573680905342, 0.9784839956315006, 0.9783199487441323, 0.9781536668275057, 0.9779983821388409, 0.9778562763301046, 0.9777210210838867, 0.9775778035163281, 0.9774390737873292, 0.9772989307245901, 0.9771565425183061, 0.9770132148220858, 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0.9664980636670173, 0.9662686017638403, 0.9660329135455574, 0.9658033522897147, 0.9655727236751724, 0.965343782721277, 0.9651079514490393, 0.9648876593651898, 0.9646666603052505, 0.9644518440146598, 0.9642473779864116, 0.964034417541067, 0.9638228149322394, 0.9636006436398837, 0.9633889917407498, 0.9631792919647791, 0.9629696804434105, 0.962770029462004, 0.9625679944272386, 0.9623682827713147, 0.9621778891208406, 0.9619901393132851, 0.9618062376128954, 0.9616259766290396, 0.9614462198985461, 0.9612627952711355, 0.9610817146835174, 0.9608959392132753, 0.9606955554682154, 0.9605044331766569, 0.9603147341573652, 0.9601203539105019, 0.9599315243986037, 0.9597266509069884, 0.959515457949925, 0.9593072505753317, 0.9590900789166856, 0.9588696934691274, 0.9586462542228903, 0.9584234622394275, 0.958201782726498, 0.9579882628625915, 0.9577812431843148, 0.9575918323129837, 0.957405852504318, 0.9572150273590522, 0.9570160922822464, 0.9568270505728019, 0.9566438294607348, 0.9564650936299884, 0.9562857577916377, 0.9560978862054401, 0.9558939920405202, 0.955684411613495, 0.9554584623037397, 0.9552321510819016, 0.954996807187403, 0.9547586402568671, 0.9545067645887038, 0.9542545816333001, 0.954007399602234, 0.9537649232673135, 0.9535120497516386, 0.9532641418933465, 0.9530081299283243, 0.9527456642257037, 0.9524886888583758, 0.9522279057479632, 0.9519801962623611, 0.9517212582094743, 0.9514597997313514, 0.951203728421824, 0.9509443737364988, 0.9506786945762492, 0.9504046773928936, 0.9501405072012504, 0.9498748182559928, 0.9495985021396359, 0.9493278550493294, 0.9490550677449313, 0.9487746265457891, 0.9484948869588958, 0.9482152052425115, 0.9479372017398722, 0.9476457808540918, 0.9473554754255046, 0.9470646692290778, 0.946767796692963, 0.9464670937526127, 0.9461646024006045, 0.9458636221411586, 0.9455566306193408, 0.9452493134598413, 0.9449304424419939, 0.9446052933848739, 0.9442766100381457, 0.9439536879049044, 0.9436361058644941, 0.9433006637798844, 0.942955939558377, 0.9426086520464716, 0.9422651162619646, 0.9419164035001663, 0.9415545142196509, 0.9412100467485824, 0.9408591861784625, 0.940528826226274, 0.9401988209509066, 0.9398764718220141, 0.9395501760422578, 0.9392282484591232, 0.9389130246670092, 0.93860653457174, 0.9382975632643129, 0.9379862166219748, 0.9376743486030519, 0.937368440601835, 0.9370549296301508, 0.9367418443383182, 0.9364332578509631, 0.9361218972141238, 0.935805466410868, 0.9354908565860288, 0.9351822102452182, 0.9348659177607546, 0.934539871188006, 0.934213708485844, 0.9338939822684609, 0.933570091349298, 0.9332624845656949, 0.9329552493557657, 0.9326567223444651, 0.9323577327041035, 0.9320683459559588, 0.9317857772726272, 0.9315047906304212, 0.9312207987242127, 0.9309264633231248, 0.930633565309053, 0.930347267366032, 0.9300411167274116, 0.9297439008618932, 0.9294552059730797, 0.9291627936557376, 0.9288777624048775, 0.9286038850107238, 0.9283285465879274, 0.9280522430443296, 0.9277738842394989, 0.9274934541956515, 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0.7998630742835893]
Les marchés obligataire et l'actif S ne sont pas corrélés car la volatilitée de l'actif S dans l'EDS qui est :
où Ct est un mouvement brownien standard independant.
On sait que la mouvement brownien du marché et le Wt qui est utilisé dans l'EDS du taux instantané rt.
or la volatilité de l'actif S et constitué aussi du movement Ct qui est indépendant du mouvement du marché wt.
Donnez le schéma d'euler :
def sig(t,x):
return 0.2* (1+ (np.sqrt(t)/(2*np.sqrt(T))) + (x/ (1+x**2)))
n =1000
r0 =3/100
alpha = 0.1
beta =1/100
theta = 5/100
T=5
pas = T/n
N=5
r=np.ones((n+1,N))*r0
integ=np.zeros((n+1,N))*r0
S0=30
TildeS=S0*np.ones((n+1,N))
S=S0*np.ones((n+1,N))
for j in range(N):
for i in range(1,n+1):
deltaWt = np.sqrt(pas)*sim.randn()
r[i,j] = r[i-1,j] + alpha*(beta-r[i-1,j])*pas+ theta*deltaWt
integ[i,j] = pas* np.sum(r[0:i-1,j]) #ici on a calculé l'intégrale pour chaque trajectoire
deltaCt=np.sqrt(pas)*sim.randn() # ici on simule le 2e mouvement brownien
# on utilise l'ancien mouvment brownien Wt
TildeS[i,j]=TildeS[i-1,j]+sig(pas*(i-1),TildeS[i-1,j])*TildeS[i-1,j]*0.25*deltaWt+(np.sqrt(15)/4)*deltaCt
#ici on simule l'actif S selon la formule déduite du tildeS
S[i,j]=TildeS[i,j]*np.exp(integ[i,j])
dates=np.linspace(0,T,n+1) # n+1 dates
graph=plt.plot(dates,S)
plt.show()
def pricingEu(k):
def sig(t,x):
return 0.2* (1+ (np.sqrt(t)/(2*np.sqrt(T))) + (x/ (1+x**2)))
def call_function(x,k):
return max(x-k,0)
n =1000
r0 =3/100
alpha = 0.1
beta =1/100
theta = 5/100
T=5
pas = T/n
N=100
r=np.ones((n+1,N))*r0
integ=np.zeros((n+1,N))*r0
ListpayoffActualise=[] #cette liste contient toutes les possibilité des payoff possible des x qui se trouve dans la dernière ligne de la matrice TildeS, ou S.
S0=30
TildeS=S0*np.ones((n+1,N))
S=S0*np.ones((n+1,N))
for j in range(N):
for i in range(1,n+1):
deltaWt = np.sqrt(pas)*sim.randn()
r[i,j] = r[i-1,j] + alpha*(beta-r[i-1,j])*pas+ theta*deltaWt
integ[i,j] = pas* np.sum(r[0:i-1,j]) #ici on a calculé l'intégrale pour chaque trajectoire
deltaCt=np.sqrt(pas)*sim.randn() # ici on simule le 2e mouvement brownien
# on utilise l'ancien mouvment brownien Wt
TildeS[i,j]=TildeS[i-1,j]+sig(pas*(i-1),TildeS[i-1,j])*TildeS[i-1,j]*0.25*deltaWt+(np.sqrt(15)/4)*deltaCt
#ici on simule l'actif S selon la formule déduite du tildeS
S[i,j]=TildeS[i,j]*np.exp(integ[i,j])
payoffActualisee =np.exp(-r*T)* call_function(S[n,j],k)
ListpayoffActualise.append(payoffActualisee)
V0Approx=np.mean(ListpayoffActualise)
print("pour K = ",k)
print("La valeur apoproximative du prix V0 de l'option Europénne: ", V0Approx)
print("------------------")
def pricingAsiatique(k):
def sig(t,x):
return 0.2* (1+ (np.sqrt(t)/(2*np.sqrt(T))) + (x/ (1+x**2)))
def put_as(x,k):
return max(k-x,0)
n =1000
r0 =3/100
alpha = 0.1
beta =1/100
theta = 5/100
T=5
pas = T/n
N=10
r=np.ones((n+1,N))*r0
integ=np.zeros((n+1,N))*r0
PayoffAct=[]
MoyS=[]
S0=30
TildeS=S0*np.ones((n+1,N))
S=S0*np.ones((n+1,N))
for j in range(N):
for i in range(1,n+1):
deltaWt = np.sqrt(pas)*sim.randn()
r[i,j] = r[i-1,j] + alpha*(beta-r[i-1,j])*pas+ theta*deltaWt
integ[i,j] = pas* np.sum(r[0:i-1,j]) #ici on a calculé l'intégrale pour chaque trajectoire
deltaCt=np.sqrt(pas)*sim.randn() # ici on simule le 2e mouvement brownien
# on utilise l'ancien mouvment brownien Wt
TildeS[i,j]=TildeS[i-1,j]+sig(pas*(i-1),TildeS[i-1,j])*TildeS[i-1,j]*0.25*deltaWt+(np.sqrt(15)/4)*deltaCt
#ici on simule l'actif S selon la formule déduite du tildeS
S[i,j]=TildeS[i,j]*np.exp(integ[i,j])
MoyS.append(np.mean(S[:,j]))
aux=put_as(MoyS[-1],k)
aux1=aux*np.exp(-integ[n,j])
PayoffAct.append(aux1)
V0Approx=np.mean(PayoffAct)
print("pour K = ",k)
print("La valeur apoproximative du prix V0 de l'option Asiatique : ", V0Approx)
print("------------------")
pricingEu(5)
pricingEu(19)
pricingEu(29)
pricingEu(31)
pricingEu(35)
pricingEu(60)
pricingEu(150)
pour K = 5 La valeur apoproximative du prix V0 de l'option Europénne: 26.36580410691194 ------------------ pour K = 19 La valeur apoproximative du prix V0 de l'option Europénne: 14.747737232857975 ------------------ pour K = 29 La valeur apoproximative du prix V0 de l'option Europénne: 7.774640420305349 ------------------ pour K = 31 La valeur apoproximative du prix V0 de l'option Europénne: 8.140216742637497 ------------------ pour K = 35 La valeur apoproximative du prix V0 de l'option Europénne: 7.5713407679087394 ------------------ pour K = 60 La valeur apoproximative du prix V0 de l'option Europénne: 1.3703515046106738 ------------------ pour K = 150 La valeur apoproximative du prix V0 de l'option Europénne: 0.0 ------------------
pricingAsiatique(5)
pricingAsiatique(19)
pricingAsiatique(29)
pricingAsiatique(31)
pricingAsiatique(35)
pricingAsiatique(60)
pricingAsiatique(150)
pour K = 5 La valeur apoproximative du prix V0 de l'option Asiatique : 0.0 ------------------ pour K = 19 La valeur apoproximative du prix V0 de l'option Asiatique : 0.0 ------------------ pour K = 29 La valeur apoproximative du prix V0 de l'option Asiatique : 0.7378871368446392 ------------------ pour K = 31 La valeur apoproximative du prix V0 de l'option Asiatique : 0.6208763500875791 ------------------ pour K = 35 La valeur apoproximative du prix V0 de l'option Asiatique : 4.0608439556842955 ------------------ pour K = 60 La valeur apoproximative du prix V0 de l'option Asiatique : 21.73099877725498 ------------------ pour K = 150 La valeur apoproximative du prix V0 de l'option Asiatique : 111.5483005383735 ------------------